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Q.

A particle is moving with uniform speed along the circumference of a circle of radius R under the action of a central fictitious force F which is inversely proportional to R³ . Its time period of revolution will be given by :

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a

$T \propto R^{\frac{3}{2}}$

b

$T \propto R^{2}$

c

$T \propto R^{\frac{4}{3}}$

d

$T \propto R^{\frac{5}{2}}$

answer is C.

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Detailed Solution

We have given $F \propto \frac{1}{R^{3}} \Rightarrow F = \frac{K}{R^{3}}$
Here K is a constant. This force will provide the required centripetal force to the particle for revolution.
$\frac{{mv}^{2}}{R} = \frac{K}{R^{3}} \Rightarrow v \propto \frac{1}{R}$

Time period of revolution, $T = \frac{2 πR}{v} ∴ T \propto R^{2}$

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A particle is moving with uniform speed along the circumference of a circle of radius R under the action of a central fictitious force F which is inversely proportional to R3 . Its time period of revolution will be given by :